EVT, Rolle's, IVT, and MVT Theorems

EVT (Extreme Value Theorem)

If f is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.

Rolle's Theorem

Let f be continuous on closed interval [a,b] and differentiable on open interval (a,b). If f(a)=f(b) then there is at least one number c on (a,b) such that f'(x)=0.

IVT (Intermediate Value Theorem)

If f is continuous on closed interval [a, b], and k is is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c)=k

*can use with speed

MVT (Mean Value Theorem)

If f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a number c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a).

*can use with speed

If you're given a function and its interval and asked to find a secant line to that and a tangent line to the function in the interval that is parallel to the secant, what do you do?

1. Find the points by looking at your ends of the interval and plugging those x's in to the function to get your y's

2. Find the average slope on that interval using those two points, which will give you the slope to use with point slope formula (this is all to find secant)

3. Now that you've found the slope, you can use mean value theorem (MVT) and the slope you just found equal to the derivative of whatever point will eventually give you the tangent line

4. Plug in the derivative of the original function for this and solve for x

5. Plug these values of x in and find the y's the go with them

6. Use each coordinate and the slope with the point slope formula (this is all to find tangent)